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What is running in the file system? Since you say you have attempted tar
and cpio I assume you have a downtime window but would like to keep is
short. It would seem that given the circumstances you have 2 options:

1. Take the downtime and run several copy commands at the same time. Since
you have indicated you have files over 2GB rsync will be slow so you'll
have to take a downtime. Do some test runs to get idea of the transfer
rate, calculate the downtime, add some padding, and take that amount of
downtime.

2. If you are running a database in this file system you can put the
database in backup mode and copy over the archive logs. I am assuming
Oracle here.

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